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A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem

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Abstract

We propose a continuous interior penalty finite element method designed for a third-order singularly perturbed problem. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Moreover, we show numerical experiments which support our theoretical findings.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000, Novi Sad, Serbia

    Helena Zarin

  2. Institute of Numerical Mathematics, Technical University of Dresden, 01062, Dresden, Germany

    Hans-Görg Roos

  3. Department for Fundamental Disciplines, Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000, Novi Sad, Serbia

    Ljiljana Teofanov

Authors
  1. Helena Zarin
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  2. Hans-Görg Roos
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  3. Ljiljana Teofanov
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Corresponding author

Correspondence to Helena Zarin.

Additional information

Communicated by Jose Alberto Cuminato.

The work of H. Zarin and the Lj. Teofanov was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia under Grant 174030.

Appendices

Appendices

1.1 Appendix 1: Proof of existence of weak solutions

First we verify the inf-sup condition

$$\begin{aligned} \mathcal{S}:=\sup _{0\ne w\in Y}\frac{B_{G}(v,w)}{|w|_{H^1(\Omega )}}\ge C\Vert v\Vert _X. \end{aligned}$$
(48)

To do that we estimate \(|v|_{H^1(\Omega )}\) and \(\varepsilon |v|_{H^2(\Omega )}\) separately. We get easily

$$\begin{aligned} |v|_{H^1(\Omega )}^2\le C B_{G}(v,v)=C \frac{B_{G}(v,v)}{|v|_{H^1(\Omega )}}|v|_{H^1(\Omega )}, \end{aligned}$$

and therefore

$$\begin{aligned} |v|_{H^1(\Omega )}\le C\,\mathcal{S}. \end{aligned}$$

For estimating \(\varepsilon |v|_{H^2(\Omega )}\) we use the possibility to estimate the \(L^2\)-norm by the sum of the \(H^{-1}\)-norm and the \(H^{-1}\)-norm of the derivative, see Steinbach (2008, Theorem 2.17). It follows

$$\begin{aligned} |v|_{H^2(\Omega )}\le C(|v''|_{H^{-1}(\Omega )}+|v'''|_{H^{-1}(\Omega )}). \end{aligned}$$

Now

$$\begin{aligned} |v''|_{H^{-1}(\Omega )}=\sup _{0\ne w\in Y} \frac{\langle v'',w\rangle }{|w|_{H^1(\Omega )}} \le |v|_{H^1(\Omega )}\le C\mathcal{S}. \end{aligned}$$

Moreover,

$$\begin{aligned} \varepsilon |v'''|_{H^{-1}(\Omega )}=\varepsilon \sup _{0\ne w\in Y} \frac{\langle v''',w\rangle }{|w|_{H^1(\Omega )}} =\varepsilon \sup _{0\ne w\in Y} \frac{(-v'',w')}{|w|_{H^1(\Omega )}}. \end{aligned}$$

Taking into account

$$\begin{aligned} \varepsilon (v'',w')=-B_{G}(v,w)+(av',w')+(bv'+cv,w), \end{aligned}$$

we get

$$\begin{aligned} \varepsilon |v'''|_{H^{-1}(\Omega )}\le \sup _{0\ne w\in Y}\frac{B_{G}(v,w)}{|w|_{H^1(\Omega )}} +\sup _{0\ne w\in Y}\frac{|A(v,w)|}{|w|_{H^1(\Omega )}}\le C\,\mathcal{S}. \end{aligned}$$

Consequently, we have proved (48).

Finally we consider the equation

$$\begin{aligned} -\varepsilon (v'',w')+A(v,w)=0. \end{aligned}$$

Taking \(v\in C_0^\infty (\Omega )\), from

$$\begin{aligned} \varepsilon \langle w'',v'\rangle =-A(v,w) \end{aligned}$$

we conclude that w belongs to the space \(H^2(\Omega )\). Moreover, it satisfies the homogenous adjoint problem

$$\begin{aligned} \varepsilon (w'',v')+A(w,v)=0 \end{aligned}$$
(49)

with the additional boundary condition \(w'(0)=0\). Setting \(w=v\) in (49), we get

$$\begin{aligned} \frac{\varepsilon }{2}(w')^2(1)+\min \{\alpha , \gamma \}\Vert w\Vert ^2_{H^1(\Omega )}\le 0, \end{aligned}$$

thus \(w=0\).

Now, the unique solvability of the Galerkin formulation (5)–(6) follows and moreover, stability with respect to the norm in X.

1.2 Appendix 2: Proof of Lemma 2

On an interval \(I_i\subset \Omega _f\), \(i=N/2+1,\ldots ,N\), starting from the standard interpolation estimate (23) and (4), we first have

$$\begin{aligned} |E-E^I|^2_{H^q(I_i)}&\le Ch_i^{2(k+1-q)}\Vert E^{(k+1)}\Vert ^2_{L^2(I_i)} \le C\varepsilon ^{-2k}h_i^{2(k+1-q)}\int _{I_i}\mathrm{e}^{-2(1-x)/\varepsilon }\,\mathrm{d}x \nonumber \\&= C\varepsilon ^{-2k+1}h_i^{2(k+1-q)}\left( \mathrm{e}^{-2(1-x_i)/\varepsilon }-\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\right) . \end{aligned}$$
(50)

From (21) and the properties of the mesh-generating and mesh-characterizing functions, we get

$$\begin{aligned} \sinh (h_i/\varepsilon )&\le Ch_i/\varepsilon =C(\phi (t_{i-1})-\phi (t_i))=C\int _{t_i}^{t_{i-1}}\phi '(t)\,\mathrm{d}t =C\int _{t_{i-1}}^{t_i}\frac{\psi '(t)}{\psi (t)}\,\mathrm{d}t \\&\le C\psi (t_{i-1})^{-1}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t = C\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t, \end{aligned}$$

in order to derive

$$\begin{aligned} \mathrm{e}^{-2(1-x_i)/\varepsilon }-\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }&= \mathrm{e}^{-2(1-x_{i-1})/\varepsilon } (\mathrm{e}^{2h_i/\varepsilon }-1) \\&= 2\mathrm{e}^{-2(1-x_{i-1})/\varepsilon } \mathrm{e}^{h_i/\varepsilon }\sinh (h_i/\varepsilon ) \\&\le C\mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t. \end{aligned}$$

Applying the last inequality into (50) gives us

$$\begin{aligned} |E-E^I|^2_{H^q(I_i)}&\le C\varepsilon ^{-2k+1}h_i^{2(k+1-q)} \mathrm{e}^{-2(1-x_{i-1})/\varepsilon }\mathrm{e}^{(1-x_{i-1})/(\tau \varepsilon )}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)} \mathrm{e}^{\left( \frac{2(k+1-q)}{\tau \varepsilon }-\frac{2}{\varepsilon }+\frac{1}{\tau \varepsilon }\right) (1-x_{i-1})} \int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)}\int _{t_{i-1}}^{t_i}\psi '(t)\,\mathrm{d}t. \end{aligned}$$

Here we have used (21) as well as

$$\begin{aligned} \frac{2(k+1-q)}{\tau \varepsilon }-\frac{2}{\varepsilon }+\frac{1}{\tau \varepsilon } =\frac{2k+3-2\tau -2q}{\tau \varepsilon }\le 0 \qquad \text{ for }\quad \tau \ge k+3/2. \end{aligned}$$

Now taking sum over all intervals from the fine part of the mesh we get

$$\begin{aligned} |E-E^I|^2_{H^q(\Omega _f)}&= \sum _{i=N/2+1}^N|E-E^I|^2_{H^q(I_i)} \nonumber \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)}\int _{1/2}^{1}\psi '(t)\,\mathrm{d}t \nonumber \\&\le C\varepsilon ^{-2q+3}(N^{-1}\max |\psi '|)^{2(k+1-q)} \end{aligned}$$

due to

$$\begin{aligned} \int _{1/2}^{1}\psi '(t)\,\mathrm{d}t=\psi (1)-\psi (1/2)=1-N^{-1}. \end{aligned}$$

Thus, we have proved (28).

Using similar arguments we derive

$$\begin{aligned} \Vert (E-E^I)^{(q)}\Vert _{L^\infty (\Omega _f)}&\le C\max _{i=N/2+1,\ldots ,N}h_i^{k+1-q}\Vert E^{(k+1)}\Vert _{L^\infty (I_i)} \nonumber \\&\le C\varepsilon ^{-q+1}(N^{-1}\max |\psi '|)^{k+1-q}\max _{i=N/2+1,\ldots ,N} \mathrm{e}^{\frac{(k+1-q)(1-x_{i-1})}{\tau \varepsilon }}\mathrm{e}^{-\frac{1-x_i}{\varepsilon }} \nonumber \\&\le C\varepsilon ^{-q+1}(N^{-1}\max |\psi '|)^{k+1-q}, \end{aligned}$$

since

$$\begin{aligned}&{\exp \left( \frac{(k+1-q)(1-x_{i-1})}{\tau \varepsilon }-\frac{1-x_i}{\varepsilon }\right) } \\&\qquad = \exp \left( \left( \frac{k+1-q}{\tau \varepsilon }-\frac{1}{\varepsilon }\right) (1-x_{i-1})+\frac{h_i}{\varepsilon }\right) \\&\qquad \le C\exp \left( \frac{k+1-\tau -q}{\tau \varepsilon }(1-x_{i-1})\right) \le C. \end{aligned}$$

Hence, the inequality (29) is also proven.

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Zarin, H., Roos, HG. & Teofanov, L. A continuous interior penalty finite element method for a third-order singularly perturbed boundary value problem. Comp. Appl. Math. 37, 175–190 (2018). https://doi.org/10.1007/s40314-016-0339-3

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